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Properties

Label 2-2100-35.3-c1-0-11
Degree $2$
Conductor $2100$
Sign $-0.0896 - 0.995i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)3-s + (0.920 + 2.48i)7-s + (−0.866 + 0.499i)9-s + (1.65 − 2.87i)11-s + (−1.05 + 1.05i)13-s + (3.20 − 0.858i)17-s + (1.10 + 1.91i)19-s + (−2.15 + 1.53i)21-s + (−1.83 + 6.83i)23-s + (−0.707 − 0.707i)27-s + (1.56 + 0.902i)31-s + (3.20 + 0.858i)33-s + (1.67 + 0.448i)37-s + (−1.29 − 0.747i)39-s − 1.33i·41-s + ⋯
L(s)  = 1  + (0.149 + 0.557i)3-s + (0.348 + 0.937i)7-s + (−0.288 + 0.166i)9-s + (0.499 − 0.865i)11-s + (−0.293 + 0.293i)13-s + (0.776 − 0.208i)17-s + (0.253 + 0.438i)19-s + (−0.470 + 0.334i)21-s + (−0.381 + 1.42i)23-s + (−0.136 − 0.136i)27-s + (0.280 + 0.162i)31-s + (0.557 + 0.149i)33-s + (0.275 + 0.0736i)37-s + (−0.207 − 0.119i)39-s − 0.207i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0896 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0896 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0896 - 0.995i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.0896 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.833873830\)
\(L(\frac12)\) \(\approx\) \(1.833873830\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-0.920 - 2.48i)T \)
good11 \( 1 + (-1.65 + 2.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.05 - 1.05i)T - 13iT^{2} \)
17 \( 1 + (-3.20 + 0.858i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.10 - 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.83 - 6.83i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-1.56 - 0.902i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.67 - 0.448i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.33iT - 41T^{2} \)
43 \( 1 + (-6.84 - 6.84i)T + 43iT^{2} \)
47 \( 1 + (-1.11 + 4.17i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.97 + 0.797i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.665 - 1.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.62 - 2.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.570 + 2.13i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.98T + 71T^{2} \)
73 \( 1 + (-2.74 - 10.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (13.8 - 7.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 + (-2.87 - 4.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.53 + 3.53i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.455949283907909588350911889497, −8.529319289169474352774266989408, −7.990265643312514042745764973041, −7.01219388082305020128573869751, −5.76734709035358370689212747462, −5.57033267587354664298191717685, −4.40846831209197514494081305130, −3.50033235252997306773511230465, −2.64522077513222246568230351269, −1.36091022557556894881764240098, 0.68343513334805449822627490125, 1.80710498748152424404334551923, 2.90549682690314874872130817040, 4.06924396835675692771393662937, 4.72405552525602613049988519943, 5.84056559900492481032806878134, 6.73424155829930886886665062221, 7.38336734470178067573544052271, 7.942708678307486829096747875135, 8.819405487476045731069108469252

Graph of the $Z$-function along the critical line